If somebody would show how to derive the matrix operation that correctly transforms the curvilinear coordinates into the cartesian coordinates, then my question will become resolved. It would also be helpful to indicate precisely where there exists error(s) in my attempted derivation.

There exists no linear transformation to map the cylindrical coordinates $\mathbf r$ to their cartesian image $\mathbf x (\mathbf r )$. The person who made the original post asked for "the matrix operation that correctly transforms the curvilinear coordinates into the cartesian coordinates", which is obviously absurd.